Infinite products are an attractive part of real analysis which has fallen out of many syllabuses. I am concerned here only with infinite products in which the factors are between 0 and 1. The partial products are positive and decreasing, and so tend to a limit; the product is said to *converge* if the limit is non-zero, or *diverge* if it is zero. (Take logarithms to see why.)

For example, the infinite product ∏(1−1/*n*) (over all *n*≥2) diverges. Yesterday, in the course of a mistaken calculation (I was calculating the wrong thing), I noticed that ∏(1−1/*n*^{2}) converges to 1/2.

**Problem:** What is ∏(1−1/*n*^{k}) for *k*≥3?

I would be quite happy to be told that this is well known! As I said, I don’t actually need the answer …

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## About Peter Cameron

I count all the things that need to be counted.

Equation (20) in

http://mathworld.wolfram.com/InfiniteProduct.html

I know nothing about infinite products. I only had a short look at this MathWorld article.

Thank you!